Evaluation of the path integral for flow through random porous media.

We present a path integral formulation of Darcy's equation in one dimension with random permeability described by a correlated multivariate lognormal distribution. This path integral is evaluated with the Markov chain Monte Carlo method to obtain pressure distributions, which are shown to agree with the solutions of the corresponding stochastic differential equation for Dirichlet and Neumann boundary conditions. The extension of our approach to flow through random media in two and three dimensions is discussed.

Hierarchical coarse-graining transform.

We present a hierarchical transform that can be applied to Laplace-like differential equations such as Darcy's equation for single-phase flow in a porous medium. A finite-difference discretization scheme is used to set the equation in the form of an eigenvalue problem. Within the formalism suggested, the pressure field is decomposed into an average value and fluctuations of different kinds and at different scales. The application of the transform to the equation allows us to calculate the unknown pressure with a varying level of detail. A procedure is suggested to localize important features in the pressure field based only on the fine-scale permeability, and hence we develop a form of adaptive coarse graining. The formalism and method are described and demonstrated using two synthetic toy problems.

Structural characterization and statistical properties of two-dimensional granular systems.

A recently developed method is used for the analysis of structures of planar disordered granular assemblies. Within this method, the assemblies are partitioned into volume elements associated either with grains or with more basic elements called quadrons. Our first aim is to compare the relative usefulness of description by quadrons or by grains for entropic characterization. The second aim is to use the method to gain better understanding of the different roles of friction and grain shape and size distributions in determining the disordered structure. Our third aim is to quantify the statistics of basic volumes used for the entropic analysis. We report the following results. (1) Quadrons are more useful than grains as basic ''quasiparticles'' for the entropic formalism. (2) Both grain and quadron volume distributions show nontrivial peaks and shoulders. These can be understood only in the context of the quadrons in terms of particular conditional distributions. (3) Increasing friction increases the mean cell size, as expected, but does not affect the conditional distributions, which is explained on a fundamental level. We conclude that grain size and shape distributions determine the conditional distributions, while their relative weights are dominated by friction and by the pack formation process. This separates sharply the different roles that friction and grain shape distributions play. (4) The analysis of the quadron volumes shows that Gamma distributions, which are accepted to describe foamlike structures well, are too simplistic for general granular systems. (5) A range of quantitative results is obtained for the ''density of states'' of quadron and grain volumes and calculations of expectation values of structural properties are demonstrated. The structural characteristics of granular systems are compared with numerically generated foamlike Dirichlet-Voronoi constructions.

Effect of anisotropy on finite-size scaling in percolation theory.

We investigate the effects of anisotropy on finite-size scaling of site percolation in two dimensions. We consider a lattice of size n(x) x n(y). We define an aspect ratio omega=n(x)/n(y) and consider the mean connected fraction P (averaged over the realizations) as a function of the site occupancy probability (p), the system size (n(x)), and this aspect ratio. It is clear that there is an easy direction for percolation, which is in the short direction (i.e., y if omega>1) and a difficult direction which is along the long axis. We define an apparent percolation threshold in each direction as the value of p when 50% of realizations connect in that direction. We show that standard finite-size scaling applies if we use this apparent threshold. We also find a finite-size scaling for the fluctuations about this mean connected fraction.

Postbreakthrough behavior in flow through porous media.

We numerically simulate the traveling time of a tracer in convective flow between two points (injection and extraction) separated by a distance r in a model of porous media, d=2 percolation. We calculate and analyze the traveling time probability density function for two values of the fraction of connecting bonds p: the homogeneous case p=1 and the inhomogeneous critical threshold case p=p(c). We analyze both constant current and constant pressure conditions at p=p(c). The homogeneous p=1 case serves as a comparison base for the more complicated p=p(c) situation. We find several regions in the probability density of the traveling times for the homogeneous case (p=1) and also for the critical case (p=p(c)) for both constant pressure and constant current conditions. For constant pressure, the first region I(P) corresponds to the short times before the flow breakthrough occurs, when the probability distribution is strictly zero. The second region II(P) corresponds to numerous fast flow lines reaching the extraction point, with the probability distribution reaching its maximum. The third region III(P) corresponds to intermediate times and is characterized by a power-law decay. The fourth region IV(P) corresponds to very long traveling times, and is characterized by a different power-law decaying tail. The power-law characterizing region IV(P) is related to the multifractal properties of flow in percolation, and an expression for its dependence on the system size L is presented. The constant current behavior is different from the constant pressure behavior, and can be related analytically to the constant pressure case. We present theoretical arguments for the values of the exponents characterizing each region and crossover times. Our results are summarized in two scaling assumptions for the traveling time probability density; one for constant pressure and one for constant current. We also present the production curve associated with the probability of traveling times, which is of interest to oil recovery.